what are the assumptions of linear regression?
Linear regression assumes that…
1. The relationship between X and Y is linear
2. Y is distributed normally at each value of X
3. The variance of Y at every value of X is the same (homogeneity of variances)
4. The observations are independent
what are some other types of multivariate regression?
Multiple linear regression is for normally distributed outcomes
Logistic regression is for binary outcomes
Cox proportional hazards regression is used when time-to-event is the outcome
What is a risk free asset?
DEFINITION: an asset whose terminal value is certain
An investment with NO risk
An asset with zero variance
Zero correlation with all other risky assets
Provides the risk-free rate of return (RFR)
does a risk free asset exit?
Given the conditions, what qualifies?
a U.S. Treasury security with a maturity matching the investor’s horizon
what is the relationship when combining a risk free asset with a risky portfolio?
Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.
Thus, the existence of a risk-free asset adds value to investors by expanding the set of portfolios available to them.
what is the portfolio separation theory?
Under the portfolio separation theory, the ideal risky portfolio in which an investor should invest is the same (P*), regardless of how aggressive or risk averse the investor is.
I.e., the point on the Markowitz efficient frontier at which the investor will invest is independent of the investor’s risk preferences.
Where risk preferences are reflected is in terms of how much of his or her portfolio is allocated to P* and how much is invested in the risk-free asset.
what are the two separation decisions for investors to make?
- The investment decision
Which portfolio on the Markowitz efficient
frontier to choose?This is determined by the point of tangency
between the Markowitz efficient frontier and
a line extending from the risk-free rateThis leads to the choice of portfolio P* as the
optimal risky portfolio for the investor,
regardless of the investor’s risk preferences - The financing decision
This is where risk preferences come into the
pictureIf the investor is more risk averse, he or she
will put part of his or her money in P* and
the rest in Treasury bonds (this is known as a
lending portfolio, because the rest of the
investor’s money is lent to the federal
government)If the investor is more aggressive, he or she
will leverage up his or her holdings and
invest in P* on margin by borrowing at the
risk-free rate (this is known as a borrowing
portfolio)
What are the general implications for security prices if investors act the way Markowitz portfolio theory and portfolio separation theory say they should? If such theories hold, what would equilibrium in the capital markets entail?
Capital market theory extends portfolio theory and develops a model for pricing all risky assets
The capital asset pricing model (CAPM) will allow you to determine the required rate of return (for use in discounting future cash flows) for any risky asset
What are the assumptions of capital market theory?
- All investors are Markowitz mean-variance efficient investors who want to target points on the efficient frontier.
- Investors can borrow or lend any amount of money at the risk-free rate of return (RFR).
- All investors have homogeneous expectations; that is, investors have identical estimates for the probability distributions of future rates of return.
- All investors have the same one-period time horizon such as one-month, six months, or one year.
- Capital markets are “frictionless,”
- Capital markets are in equilibrium.
what is systemic risk?
Only systematic risk remains in the market portfolio
Systematic risk is the variability in all risky assets caused by macroeconomic variables
Systematic risk can be measured by the standard deviation of returns of the market portfolio and can (and does) change over time
what are some example of macroeconomic factors that affect systematic risk?
Variability in growth of money supply
Interest rate volatility
Variability in:
industrial production
corporate earnings
and cash flow
what is the capital market line (CML)?
Describes the risk / return relationship for well-diversified portfolios (idiosyncratic risk has been diversified away).
Portfolio standard deviation (Q) is the relevant measure of risk, and the portfolio’s expected return (E(RQ)) will be a direct linear function of its risk
To obtain higher expected returns, must accept higher risk.
What is the relevant measure of risk for an individual security when it is held as part of a well diversified portfolio (i.e., the Market portfolio, M)?
The Security Market Line describes the risk / return relationship for an individual security.
Also applies to non-diversified portfolios or any other holding for which the total risk may include some diversifiable or idiosyncratic risk.
how do determine the expected rate of return for a risky asset?
The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset
The risk premium is determined by the systematic risk of the asset () and the prevailing market risk premium (RM-RFR)
In equilibrium, to obtain higher expected returns, investors must accept higher “covariance” risk
In equilibrium, investors receive no compensation for diversifiable (non-systematic or idiosyncratic) risk
how to identify undervalued and overvalued assets?
Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment
Independent estimates of return for the securities provide price and dividend outlooks
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Class 1: Financial Risk and Modeling18
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Class 2: Profit Maximization and Risk Minimization as Dual Problems6
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Class 3: Credit Risk26
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Class 4: Market Risk34
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Class 6: ESG14
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Class 9: Linear Models15
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Class 12: Probability and Statistics for Financial Modeling16
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Class 13: Non-Linear Models15
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Final Prep18
